Meghna Nandi, Kristin L. Schreiber, Marc O. Martel, Marise Cornelius, Claudia M. Campbell, Jennifer A. Haythornthwaite, Michael T. Smith, John Wright, Linda S. Aglio, Gary R. Strichartz, Robert R. Edwards,
Knee osteoarthritis (OA) is among the most common and disabling persistent pain conditions, with increasing prevalence in the developed world, and affects women to a greater degree than men. In the USA, the growth of knee OA has been paralleled by an increase in rates of total knee arthroplasty (TKA), a surgical treatment option for late-stage knee OA. While TKA outcomes are generally good, postoperative trajectories of pain vary widely, with some patients reporting a complete absence of pain, but ...
Tópico(s): Anesthesia and Pain Management
2019 - BioMed Central | Biology of Sex Differences
Nantthasorn Zinboonyahgoon, Kamen Vlassakov, Philipp Lirk, Tara Spivey, Tari A. King, Laura S. Dominici, Mehra Golshan, Gary R. Strichartz, Robert R. Edwards, Kristin L. Schreiber,
BackgroundPrevious studies suggest that truncal regional anaesthesia (TRA), including techniques such as paravertebral block, may contribute significantly to analgesia after mastectomy. However, the severity and impact of postoperative pain varies markedly amongst individuals, making the identification of patients who would benefit most from TRA a potentially important step toward personalised perioperative care.MethodsIn this prospective observational study, mastectomy patients (n=122) were recruited ...
Tópico(s): Dental Anxiety and Anesthesia Techniques
2019 - Elsevier BV | British Journal of Anaesthesia
Sarah Constantin, Robert Strichartz, Miles H. Wheeler,
We study the spectral decomposition of the Laplacian on a family of fractals $\mathcal{VS}_n$ that includes the Vicsek set for $n=2$, extending earlier research on the Sierpinski Gasket. We implement an algorithm [23] for spectral decimation of eigenfunctions of the Laplacian, and explicitly compute these eigenfunctions and some of their properties. We give an algorithm for computing inner products of eigenfunctions. We explicitly compute solutions to the heat equation and wave equation for Neumann ...
Tópico(s): Numerical methods in inverse problems
2010 - American Institute of Mathematical Sciences | Communications on Pure &Applied Analysis
Kevin Romeo, Benjamin Steinhurst,
... The authors thank Alexander Teplyaev, Daniel Ford, and Robert Strichartz for their support and feedback throughout the production ...
Tópico(s): Random Matrices and Applications
2009 - Taylor & Francis | Complex Variables and Elliptic Equations
Tópico(s): Matrix Theory and Algorithms
1965 - Mathematical Sciences Publishers | Pacific Journal of Mathematics
... and his coinventors (Dr. Daniel Kohane, Dr. Gary Strichartz, Dr. Robert Langer) hold issued patents on multiple approaches to ...
Tópico(s): Medical History and Innovations
2015 - Lippincott Williams & Wilkins | Anesthesia & Analgesia
Skye Aaron, Zach Conn, Robert Strichartz, Yu Han,
We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and 3-dimensional Sierpinski gasket (SG$^3$), but the method is expected to be effective for many PCF fractals, and also infinitely ramified fractals such as the Sierpinski carpet (SC). Our approach is to construct k-forms and de Rham differential operators $d$ and $\delta$ for a sequence of graphs approximating the fractal, and then pass to the limit with ...
Tópico(s): Topological and Geometric Data Analysis
2013 - American Institute of Mathematical Sciences | Communications on Pure &Applied Analysis
Tópico(s): Medical Imaging Techniques and Applications
1982 - Taylor & Francis | American Mathematical Monthly
Robert Ravier, Robert S. Strichartz,
In the case of some fractals, sampling with average values on cells is more natural than sampling on points. In this paper we investigate this method of sampling on SG and SG $$_{3}$$ . In the former, we show that the cell graph approximations have the spectral decimation property and prove an analog of the Shannon sampling theorem. We also investigate the numerical properties of these sampling functions and make conjectures that allow us to look at sampling on infinite blowups of SG. In the case of SG $$_{ ...
Tópico(s): Mathematical Analysis and Transform Methods
2016 - Springer Science+Business Media | Constructive Approximation
Matthew Begué, Tristan Kalloniatis, Robert S. Strichartz,
Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate solutions to boundary value problems. As a result we have a wealth of data concerning harmonic functions with prescribed boundary values, and eigenfunctions of the Laplacian with either Neumann or Dirichlet boundary conditions. We will present some of this data and discuss ...
Tópico(s): advanced mathematical theories
2013 - World Scientific | Fractals
Marius Ionescu, Luke G. Rogers, Robert S. Strichartz,
We define and study pseudo-differential operators on a class of fractals that include post-critically finite (p.c.f.) self-similar sets and Sierpiński carpets. Using sub-Gaussian estimates for the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to products of fractals. While our results are applicable to a larger class of metric measure spaces with Laplacian, we use them to study elliptic, ...
Tópico(s): Advanced Mathematical Modeling in Engineering
2013 - Royal Spanish Mathematical Society | Revista Matemática Iberoamericana
Taryn C. Flock, Robert S. Strichartz,
We describe families of Laplacians on Julia Sets $\mathcal {J}_c$ for quadratic polynomials $P(z)=z^2+c$ in the spirit of Kigami's construction of Laplacians on p.c.f. self-similar fractals. We consider an infinite family of Julia sets for $c$ in the interior of a bulb in the Mandelbrot set that includes the basilica and the Douady rabbit. We use the external ray parametrization of the Julia set which represents the Julia set as a circle with some points identified. There is a one-dimensional space of $ ...
Tópico(s): Theoretical and Computational Physics
2012 - American Mathematical Society | Transactions of the American Mathematical Society
Tarik Aougab, Stella Chuyue Dong, Robert S. Strichartz,
This paper continues the work started in [4] to construct $P$-invariant Laplacians on the Julia sets of $P(z) = z^2 + c$ for $c$ in the interior of the Mandelbrot set, and to study the spectra of these Laplacians numerically. We are able to deal with a larger class of Julia sets and give a systematic method that reduces the construction of a $P$-invariant energy to the solution of nonlinear finite dimensional eigenvalue problem. We give the complete details for three examples, a dendrite, the airplane, and ...
Tópico(s): Advanced Mathematical Modeling in Engineering
2012 - American Institute of Mathematical Sciences | Communications on Pure &Applied Analysis
Robert S. Strichartz, Alexander Teplyaev,
A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpinski gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpinski gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of ...
Tópico(s): Theoretical and Computational Physics
2012 - Springer Science+Business Media | Journal d Analyse Mathématique
Steven Heilman, Philip Owrutsky, Robert S. Strichartz,
We study experimentally systems of orthogonal polynomials with respect to self-similar measures. When the support of the measure is a Cantor set, we observe some interesting properties of the polynomials, both on the Cantor set and in the gaps of the Cantor set. We introduce an effective method to visualize the graph of a function on a Cantor set. We suggest a new perspective, based on the theory of dynamical systems, for studying families Pn (x) of orthogonal functions as functions of n for fixed ...
Tópico(s): Quantum chaos and dynamical systems
2011 - Taylor & Francis | Experimental Mathematics
Marius Ionescu, Erin P. J. Pearse, Luke G. Rogers, Huo-Jun Ruan, Robert S. Strichartz,
For the Laplacian Δ \Delta defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions and also with Neumann boundary conditions. That is, we construct a symmetric function G ( λ ) G^{(\lambda )} which solves ( λ I − Δ ) − 1 f ( x ) = ∫ G ( λ ) ( x , y ) f ( y ) d μ ( y ) (\lambda \mathbb {I} - \Delta )^{-1} f(x) = \int G^{(\lambda )}(x,y) f(y) \, d\mu (y) . The method is similar to Kigami’s construction of the Green kernel and G ( λ ) G^{(\lambda )} is expressed as a sum of scaled ...
Tópico(s): Advanced Mathematical Theories and Applications
2010 - American Mathematical Society | Transactions of the American Mathematical Society
Tyrus Berry, Steven Heilman, Robert S. Strichartz,
We present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate K from the outside. The method allows a numerical approximation of eigenvalues and eigenfunctions for lower portions of the spectrum. We present experimental evidence that the method works by looking at examples in which the spectrum of the fractal Laplacian is known (the unit interval ...
Tópico(s): Theoretical and Computational Physics
2009 - Taylor & Francis | Experimental Mathematics
Shawn Drenning, Robert S. Strichartz,
We give a complete description of the Dirichlet and Neumann spectra of the Laplacian on a class of homogeneous hierarchical fractals introduced by Hambly. These fractals are finitely ramified but not self-similar. We use the method of spectral decimation. As applications, we show that these spectra always have infinitely many large spectral gaps, allowing for nice convergence results for eigenfunction expansions, and under certain restrictions we give a computer-assisted proof that the set of ratios ...
Tópico(s): Quantum chaos and dynamical systems
2009 - Duke University Press | Illinois Journal of Mathematics
Jessica L. DeGrado, Luke G. Rogers, Robert S. Strichartz,
We use spectral decimation to provide formulae for computing the harmonic tangents and gradients of Laplacian eigenfunctions on the Sierpinski Gasket. These formulae are given in terms of special functions that are defined as infinite products.
Tópico(s): Mathematical Analysis and Transform Methods
2008 - American Mathematical Society | Proceedings of the American Mathematical Society
Edward Fan, Zuhair U. Khandker, Robert S. Strichartz,
Tópico(s): Mathematical Analysis and Transform Methods
2008 - Springer Science+Business Media | Communications in Mathematical Physics
Adam Allan, Michael Bárány, Robert S. Strichartz,
We study spectral operators for the Kigami Laplacian on the Sierpinski gasket. These operators may be expressed as functions of the Laplacian (Dirichlet or Neumann), or as Fourier multipliers for the associated eigenfunction expansions. They include the heat operator, the wave propagator and spectral projections onto various families of eigenspaces. Our approach is both theoretical and computational. Our main result is a technical lemma, extending the method of spectral decimation of Fukushima and ...
Tópico(s): Mathematical Analysis and Transform Methods
2008 - Taylor & Francis | Complex Variables and Elliptic Equations
We study the Schrödinger operator $ H = - \Delta + V $ on theproduct of two copies of an infinite blowup of the Sierpinski gasket,where $ V$ is the analog of a Coulomb potential ($\Delta V$ is amultiple of a delta function). So $H$ is the analog of the standardHydrogen atom model in nonrelativistic quantum mechanics. Like theclassical model, we show that the essential spectrum of $H$ is thesame as for $ - \Delta $, and there is a countable discrete spectrumof negative eigenvalues.
Tópico(s): Statistical Mechanics and Entropy
2008 - American Institute of Mathematical Sciences | Communications on Pure &Applied Analysis
Brian Bockelman, Robert S. Strichartz,
We describe a finite element method based on piecewise pluriharmonic or piecewise pluribiharmonic splines to numerically approximate solutions to partial differential equations on the product SG 2 of two copies of the Sierpinski gasket. We use this method to experimentally study both elliptic equations, and a new class of operators that we call quasielliptic, which has no analog in the standard theory of pde's. The existence of these operators is based on the observation that the set of ratios of ...
Tópico(s): Algebraic and Geometric Analysis
2007 - Indiana University | Indiana University Mathematics Journal
Jonas Azzam, Michael A. Hall, Robert S. Strichartz,
On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy E φ \mathcal {E}_\varphi and conformal Laplacian Δ φ \Delta _{\varphi } for a given conformal factor φ \varphi , based on the corresponding notions in Riemannian geometry in dimension n ≠ 2 n\neq 2 . We derive a differential equation that describes the dependence of the effective resistances of E φ \mathcal {E}_\varphi on φ \varphi . We show that the spectrum of Δ φ \Delta _{\varphi } (Dirichlet or Neumann) has similar asymptotics compared ...
Tópico(s): Quantum chaos and dynamical systems
2007 - American Mathematical Society | Transactions of the American Mathematical Society
Nitsan Ben-Gal, Abby Shaw-Krauss, Robert S. Strichartz, Clint Young,
This paper continues the study of fundamental properties of elementary functions on the Sierpinski gasket (SG) related to the Laplacian defined by Kigami: harmonic functions, multiharmonic functions, and eigenfunctions of the Laplacian. We describe the possible point singularities of such functions, and we use the description at certain periodic points to motivate the definition of local derivatives at these points. We study the global behavior of eigenfunctions on all generic infinite blow-ups of ...
Tópico(s): Quantum chaos and dynamical systems
2006 - American Mathematical Society | Transactions of the American Mathematical Society
Tópico(s): Advanced Harmonic Analysis Research
2006 - Springer Science+Business Media | Journal d Analyse Mathématique
Tópico(s): Theoretical and Computational Physics
2005 - Springer Science+Business Media | Journal d Analyse Mathématique
Kasso A. Okoudjou, Robert S. Strichartz,
We use the analytic tools such as the energy, and the Laplacians defined by Kigami for a class of post-critically finite (pcf) fractals which includes the Sierpinski gasket (SG), to establish some uncertainty relations for functions defined on these fractals. Although the existence of localized eigenfunctions on some of these fractals precludes an uncertainty principle in the vein of Heisenberg’s inequality, we prove in this article that a function that is localized in space must have high energy, ...
Tópico(s): Computability, Logic, AI Algorithms
2005 - Birkhäuser | Journal of Fourier Analysis and Applications
On the Sierpinski gasket and related fractals, partial sums of Fourier series (spectral expansions of the Laplacian) converge along certain special subsequences.This is related to the existence of gaps in the spectrum.
Tópico(s): Mathematical Analysis and Transform Methods
2005 - International Press of Boston | Mathematical Research Letters
Robert A. Meyers, Robert S. Strichartz, Alexander Teplyaev,
We study not necessarily self-similar Dirichlet forms on the Sierpiński gasket that can be described as limits of compatible resistance networks on the sequence of graphs approximating the gasket.We describe the compatibility conditions in detail, and we also present an alternative description, based on just 3 conductance values and the 3-dimensional space of harmonic functions.In addition, we show how to parameterize all the Dirichlet forms by a set of independent variables.
Tópico(s): Theoretical and Computational Physics
2004 - Mathematical Sciences Publishers | Pacific Journal of Mathematics